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Ahmad Al Hanbali, Philippe Nain, Eitan Altman

To cite this version:

Ahmad Al Hanbali, Philippe Nain, Eitan Altman. Performance Evaluation of Packet Relaying in Ad

Hoc Networks. [Research Report] RR-5860, INRIA. 2006, pp.24. �inria-00070166�

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ISRN INRIA/RR--5860--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

Performance Evaluation of Packet Relaying in Ad

Hoc Networks

Ahmad Al Hanbali, Philippe Nain, and Eitan Altman

N° 5860

Mars 2006

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Ahmad Al Hanbali, Philippe Nain, and Eitan Altman

Thème COM — Systèmes communicants Projets Maestro

Rapport de recherche n° 5860 — Mars 2006 — 24 pages

Abstract: Considered is a mobile ad hoc network consisting of three types of nodes (source, destination and relay nodes) and using the two-hop relay routing protocol. Packets at relay nodes are assumed to have a limited lifetime in the network. All nodes are moving inside a bounded region according to some random mobility model. Both closed-form expressions, and asymptotic results when the number of nodes is large, are provided for a number of QoS metrics related to the packet delivery delay and the energy needed to transmit a packet from the source to its destination. Markov models and mean-field approximations are used. We also introduce and evaluate a variant of the two-hop relay protocol that limits the number of copies of a packet in the network. Our model is validated through simulations for two mobility models (random waypoint and random direction mobility models), numerical results for the two-hop relay protocols are reported, and the performance of the two-hop routing and of the epidemic routing protocols are compared.

Key-words: Ad hoc network, Two-hop relay protocol, Limited packet lifetime, Mobility model, Markovian analysis, Mean-field approximation, QoS metric.

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les Réseaux Ad Hoc

Résumé : Nous considérons un réseau ad hoc mobile constitué de trois types de nœuds: source, destination et nœud relais. Le rôle du nœud relais est de transmettre les paquets entre la source et la destination en l’absence d’une route directe entre ces derniers. Nous supposons que la durée de vie des paquets transmis par les noeuds relais est limitée. Tous les nœuds se déplacent indépendamment les uns des autres, selon le même modèle de mobilité. Nous établissons les formules de la distribution de probabilité des temps de délai pour l’envoi des paquets à la destination, ainsi que la quantité d’énergie consommée par le réseau pour l’envoi des paquets. Ces formules ont été trouvées en utilisant un modèle markovien et un modèle fluide. En plus, nous étudions l’effet sur les temps de délai de limiter la quantité d’energie à consommer par le rèseau . En comparant les résultats théoriques trouvés et les résultats des simulations, nous avons trouvé que notre modèle est précis surtout pour les petites valeurs de portée de transmission des noeuds et lorsque le nombre des noeuds est fini.

Mots-clés : réseaux ad hoc, relais des paquets, paquet de durée de vie limitée, modèles de mobilité, analyse Markovienne, évaluation de performance.

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1

Introduction

Ad hoc Networks are complex distributed systems, that are composed of wireless mobile or static nodes that can freely and dynamically self-organize. In this way they form arbitrary, and temporary “ad hoc” network topologies, allowing devices to seamlessly interconnect in areas with no pre-existing infrastructure.

In a Mobile Ad Hoc Network (MANET), since there is no fixed infrastructure and nodes are mobile, links between nodes are set up and turn down dynamically. A link between two nodes is up when these nodes are inside one another communication range, and a link is down otherwise. The es-tablishment of a route from a source node to a destination node requires the simultaneous availability of a number of links that are all up, one originating at the source node and another one ending at the destination nodes. Indeed, MANETs often experience route failures and network disconnectivity, especially when the nodes are moving frequently and the network is sparse. Grossglauser and Tse [9] have observed that mobility in MANETs can be used to increase the average network through-put. Their idea was to look at the diversity gain achieved by using the mobile nodes as relays. Their relay mechanism, called two-hop relay protocol, is simple: if there is no route between the source node and the destination node, the source node transmits its packets to all neighboring nodes (called

relay nodes) that it meets for delivery to the destination. A relay node is only allowed to send a

packet to its destination node, and it is not allowed to send the packet to another relay node, thereby justifying the name of this protocol. It was then shown in [6] that a bounded delay can be guaranteed under this relaying mechanism. The aim of these studies (see also [10]) is the scaling property of the throughput or delay as the number of nodes in the network becomes large. Our interest in the present work is in the performance of the above mentioned relaying mechanism in a network consisting of a fixed finite number of nodes.

It is important to mention that most of the studies of scaling laws of delay and throughput in wireless MANETs assume a uniform spatial distribution of nodes, which is the case, for example, when the nodes perform a symmetric Random Walk over the region of interest [6, 9], or when nodes move according to the Random Direction model [13]. In the present work, we replace this assump-tion by assuming that the inter-meeting time between two nodes, defined as the time duraassump-tion time between two consecutive points in time where these nodes meet (i.e. come within transmission range of one another), is exponentially distributed. The validity of this assumption has been discussed in [8], and its accuracy has been shown for a number of mobility models (Random Walker, Random Direction, Random Waypoint) in the case when the node transmission range is small with respect to the area where the nodes evolve. It is worth pointing out that for some of these mobility models (non-symmetric Random Walk and Random Waypoint) nodes are not uniformly distributed over the area of interest.

The objective of this paper is to study a number of quality of service (QoS) metrics bearing on the packet delivery delay and the overhead induced by the two-hop relay protocol (see Section 2). This will be done under the assumption that, unlike in [8, 15], packets at relay nodes have a limited lifetime in the network.

Another relay protocol closely related to the two-hop relay protocol is the so-called epidemic

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epidemic routing protocol a relay node is allowed to transmit a packet to any node that its meets, including another relay node. Epidemic routing decreases the delivery delay of packets at the cost of increasing the energy consumption by the network. The performance of both the two-hop relay protocol and the epidemic routing protocol will also be compared in this paper.

The rest of the paper is organized as follows: Section 2 gives a careful description of the two-hop relay protocol, sets the modeling assumptions, and defines the QoS metrics of interest (delivery delay, overhead in terms of the number of copies of a packet). In Section 3 we develop a Markovian analysis that yields closed-form expressions for these QoS metrics. In Section 4, we propose and evaluate a modification of the two-hop relay protocol, calledK-limited two-hop relay protocol,

that aims at limiting the overall energy consumption. Section 5 presents an asymptotic analysis of the QoS metrics as the number of nodes is large; this analysis uses a mean-field approximation. Validation of our model, and comparison of the performance of the two-hop relay protocol and the epidemic routing protocol are given in Section 6. Section 7 concludes the paper and suggests some research directions.

2

The MANET model

We consider the MANET model introduced in [8]. In this model the characteristics of a MANET are captured through a single parameter,1/λ , representing the expected inter-meeting time between

any pair of nodes. More precisely, there areN + 1 nodes, one source node, one destination node

andN − 1 relay nodes. Two nodes may only communicate at certain points in time, called meeting times. The time that elapses between two consecutive meeting times of a given pair of nodes is called the inter-meeting time. In [8] it is assumed that inter-meeting times are mutually independent and identically distributed (iid) random variables (rvs), with an exponential distribution with intensity

λ > 0.

Throughout this paper we address the unique scenario where the source wants to send a single packet to the destination node. To this end the source node may use the relay nodes, as explained below. In this paper, we will focus on the two-hop relay protocol [9].

The two-hop relay protocol works as follows. The source node keeps sending a copy of the packet to all nodes that it meets and that do not have a copy, including the destination node, until the destination node has received a copy of the packet. Transmissions are assumed to be instantaneous. The way the source node is notified that the destination node has received the packet, either directly from it or from a relay node, is irrelevant for the metrics that we will consider (see below).

A relay node that possesses a copy of the packet is only allowed to send it to the destination node, thereby justifying the name of this protocol (two-hop relay protocol).

In addition to the model in [8] we assume throughout this paper that each copy of the packet has a Time-To-Live (TTL). When the TTL of a copy expires then the copy is destroyed. TTLs are assumed to be iid rvs with an exponential distribution with rateµ > 0. The packet to be sent by

the source has no TTL associated with it, so that the source is always able to send a copy to another node (if the packet at the source has a TTL then there is a non-zero probability that the destination node will never receive the packet. This scenario is not considered in this paper).

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We assume that the source is ready to transmit the packet to the destination at timet = 0. The

(packet) delivery time (or delivery delay),Td, is the first time aftert = 0 when the destination node

receives the packet (or a copy of the packet).

In the following we will investigate the delivery delay, the number of copies in the system at the delivery time, and the total number of copies generated by the source before the delivery time (Section 3). The latter is related to the overhead induced by the two-hop relay protocol and, in particular, to the total energy needed to transmit the packet to the destination (Section 4).

A word on the notation: throughout 1A will designate the indicator function of any event A

(1A = 1 if A is true and 0 otherwise) and diag (a1, . . . , an) will define a n-by-n diagonal matrix

with(i, i)-entry ai.

3

Markovian Analysis

The state of the system is represented by the random variableI(t) ∈ {1, 2, . . . , N, a}, where I(t) ∈

{1, 2, . . . , N} gives the number of copies when the packet has not been delivered to the destination

(i.e. for0 ≤ t < Td) and I(t) = a for t ≥ Td. Under the assumptions made in Section 2,

{I(t), t ≥ 0} is an absorbing, finite-state, continuous-time Markov chain, with transient states {1, 2, . . . , N} and absorbing state a. Let P = [p(i, j)] be the one-step transition matrix of the

absorbing, finite-state, discrete-time Markov chain (referred to as MC from now on) embedded just before the jump times of the Markov chain {I(t), t ≥ 0}. From the transition rate diagram of Markov chain{I(t), t ≥ 0} in Figure 1 we find

p(i, i + 1) = (N − i)λ N λ + (i − 1)µ = (N − i)ρ N ρ + i − 1, i = 1, . . . , N − 1, p(i, i − 1) = (i − 1)µ N λ + (i − 1)µ = i − 1 N ρ + i − 1, i = 2, . . . , N, p(i, a) = iλ N λ + (i − 1)µ = iρ N ρ + i − 1, i = 1, . . . , N, p(i, j) = 0, otherwise, withρ := λ/µ.

The transition matrix P of the Markov chain MC can be written as

P=  Q R 0 1  ,

where Q= [p(i, j)]1≤i,j≤N, R= (p(1, a), . . . , p(N, a))T, and 0 is the row vector of dimensionN

whose all components are equal to0.

Define M= (I − Q)−1, the fundamental matrix of the absorbing Markov chain MC. The(i,

j)-entry of M, denoted bym(i, j), gives the expected number of visits to state j given that I(0) = i

[7, Chap. 11, Theorem 11.4].

The matrix M is computed in explicit form in Lemma 2 in the Appendix I, andm(i, j) is given

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λ N iλ 1 2 i (N−1) λ λ λ µ 2µ µ 2 λ λ (N−2) (N−i+1) (N−i) λ (i−1) iµ N λ (N−1) λ 2 µ (N−1) (N−2) µ λ N−1 a

Figure 1: Transition rate diagram of the Markov chain{I(t), t ≥ 0}.

We are now in position to compute the expected delivery delay, the distribution of the number of copies at the delivery instant, and the expected number of copies generated by the source.

3.1

Delivery delay

In this section we first determine Ei[Td], the expected delivery delay given that I(0) = i ∈

{1, 2, . . . , N}, from which the expected delivery delay Td= E1[Td] will follow.

Ei[Td] is the expected time before absorption starting from the transient state i. Let nij be the

number of visits to statej before absorption given that the chain starts in state i, and let Tjlbe the

sojourn time in statej at the lth visit to that state. Observe that E[nij] = m(i, j), where m(i, j) is

given in Lemma 2, and thatE[Tjl] = 1/(N λ + µ(j − 1)) for j = 1, 2, . . . , N (see Figure 1). Hence,

Ei[Td] = E   N X j=1 nij X l=1 Tjl  = N X j=1 E "nij X l=1 Tjl # = N X j=1 m(i, j)E[Tjl],

where the last equality follows from Wald’s identity, sinceni,jis independent of the rvs{Tjl}j,l.

Plugging the value found in (17) form(i, j) in the latter equation gives

Ei[Td] = −1 µ N − 1 i − 1  ρi−1 −1 N X k=1 Ψk i zkΨkτ2(Ψk)T Ψk1T, (1)

with 1T theN -dimensional column vector whose all components are equal to 1. Quantities Ψk i,τ

andzkare defined in Lemma 2 in the Appendix I.

More generally, the probability distribution ofTd, starting fromI(0) = i is given by

Pi(Td< t) = 1 −N − 1 i − 1  ρi−1 −1 N X k=1 Ψk i Ψkτ2k)TΨ k1Tezkµt.

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3.2

Number of copies in the system at delivery time

LetPi[Cd= j] be the probability that the number of copies in the network at the delivery time is j,

given there arei copies in the network at time t = 0. We assume without loss of generality that the

Markov chain MC is left-continuous so thatPi[Cd= j] = P [I(Td−) = j] (by convention I(t−) is

the state of the process MC just before timet). In words, Pi[Cd = j] is the probability that the last

visited state before absorption isj, given that the initial state is i.

If we split the absorbing statea into N absorbing states a1, . . . , aN, as shown in Figure 2, we

will not affect the dynamics of the original Markov chain before absorption. This means that the fundamental matrix of the modified absorbed Markov chain is the same as the fundamental matrix of the original absorbed Markov chain. Clearly,Pi[Cd = j] is now equal to the probability that the

modified chain is absorbed in stateaj. Letbi,ajdenote this probability. From the theory of absorbing

Markov chains, we see that [7, Chap. 11, Theorem 11.6]

bi,aj =

N

X

k=1

m(i, k)r(k, aj),

wherer(k, aj) is the one-step transition probability from state k to the absorbing state aj in the

modified Markov chain. Clearly (see Figure 2)r(k, aj) = jλ/(N λ + (j − 1)µ) = jρ/(Nρ + j − 1)

ifk = j and r(k, aj) = 0 if k 6= j. Therefore, Pi[Cd= j] = m(i, j)r(j, aj) = −jN − 1 i − 1  ρi−2 −1 N X k=1 Ψk iΨkj zkΨkτ2(Ψk)T .

Thenth-order moment of Cdis equal to (with Jn := (1, · · · , in, · · · , Nn))

Ei[Cdn] = −N − 1 i − 1  ρi−2 −1 N X k=1 Ψk i zkΨkτ2(Ψk)T ΨkJTn+1. (2)

Coming back to the original problem, the probability distribution of the number of copies at delivery time is given byP1[Cd= j], and the n-th order moment is given by E1[Cdn].

λ N iλ a1 a2 ai aN−1 aN 1 2 i (N−1) λ λ λ µ 2µ µ 2 λ (N−2) (N−i+1) (N−i) λ (i−1) iµ N λ (N−1) λ 2 µ (N−2) µ N−1 λ (N−1)λ

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3.3

Total number of copies generated by the source before delivery time

The objective is to find, Gd, the expected number of copies generated by the source before the

delivery time (or equivalently, before absorption). LetGi,jd be the number of copies generated by the source before absorption given that the chain starts in statei and that state j is the last state visited

before absorption (i.e. I(Td−) = j). Introduce Ji,j(k, k + 1) (resp. Ji,j(k + 1, k)) the number

of transitions from statek (resp. state k + 1) to state k + 1 (resp. state k) given that I(0) = i and I(Td−) = j. It is easy to see that

Ji,j(k, k + 1) = Ji,j(k + 1, k) + 1

{i≤k<j}− 1{j≤k<i}, k = 1, 2, . . . , N − 1. (3)

A copy of the packet is generated by the source each time there is a transition from statek to state k + 1 for all states k = 1, 2, . . . , N − 1. Hence,

Gi,jd =

N −1

X

k=1

Ji,j(k, k + 1).

On the other hand,nji,k, the total number of visits to statek given that I(0) = i and I(Td−) = j,

satisfies the relation

N X k=1 nji,k= N −1 X k=1 Ji,j(k, k + 1) + N −1 X k=1 Ji,j(k + 1, k) + 1.

From (3) we find that

N −1 X k=1 Ji,j(k + 1, k) = N −1 X k=1 Ji,j(k, k + 1) + i − j.

Combining the three last identities gives

Gi,jd = 1 2 "N X k=1 nji,k+ j − i − 1 # .

The expected number of copies given thatI(0) = i, denoted by Gi

d, is given by (Hint: remove the

conditioning onCd = j) Gi d= N X j=1 Gi,jd P (Cd= j) = 1 2 "N X k=1 m(i, k) + Ei[Cd] − i − 1 #

wherem(i, k) is given in Lemma 2 and Ei[Cd] is given in (2) (with n = 1).

Finally,Gd = G1d. We will see in the next section howGdcan be used to compute the overall

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4

Limiting the energy consumption

We will only consider the energy consumption due to packet transmission and decoding. Let pt

be the energy needed at the sender to transmit a packet to another node and letpr be the energy

needed at the receiver to decode a packet. The energy consumed by the source before the packet is delivered to the destinationPs= ptGd, since the source needs to generate on the averageGdcopies

of the packet before one copy reaches the destination. The energy consumed by all nodes before the delivery time is given byPd = (pt+ pd) Gd.

In this section we introduce and evaluate a new two-hop relay scheme that limits the energy con-sumption by limiting the number of copies that the source can generate before the packet reaches the destination. A similar scheme was introduced in [15] to limit the energy consumption of epidemic routing. We now assume that the source can generated at mostK copies of the packet. In the

follow-ing this scheme will be referred to as theK-limited two-hop relay protocol. Alike in the original

pro-tocol in Section 3 (corresponding toK = ∞), we will compute the expected delivery delay and the expected number of copies generated before the delivery time for theK-limited two-hop relay

pro-tocol. The behavior of theK-limited two-hop relay protocol can be modeled as a two-dimensional,

finite-state, absorbing and continuous-time Markov chain (referred to as MCK) with state(i, c),

wherei ∈ {1, 2, . . . , N} gives the number of copies in the network, and c ∈ {0, 1, . . . , K} records the total number of copies generated by the source. It is easy to see that the one-step probabil-ity transition matrix PK = [pK((i, c), ·)] of the absorbing, finite-state, discrete-time Markov chain

(referred to as MCK)) embedded just before the jump times of MCKis given by

pK((i, c), (i + 1, c + 1)) = (N − i)ρ N ρ + (i − 1), i = 1, . . . , Km, c = i − 1, . . . , K − 1, pK((i, c), (i − 1, c)) = (i − 1) N ρ + (i − 1), i = 2, . . . , Km, c = i − 1, . . . , K − 1, pK((i, c), a) = iρ N ρ + (i − 1), i = 1, . . . , Km, c = i − 1, . . . , K − 1, pK((N, c), (N − 1, c)) = (N − 1) N ρ + (N − 1)1{K≥N}, c = N − 1, . . . , K − 1, pK((N, c), a) = N ρ N ρ + (N − 1)1{K≥N}, c = N − 1, . . . , K − 1, pK((i, K), (i − 1, K) = (i − 1) iρ + (i − 1), i = 2, . . . , Km+ 1, pK((i, K), a) = iρ iρ + (i − 1), i = 1, . . . , Km+ 1, pK((i, c), (j, d)) = 0, otherwise,

withKm := min(K, N − 1), and where a is the absorbing state. If K ≤ N − 1 then the total

number of transient states isL1 := (K + 1)(K + 2)/2 whereas if K > N this number is equal to

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If we label the transient states(1, 0) as 1, (2, 1) as 2, · · · , (i, c) as (c+1)(c+2)2 − i + 1 for c ≤ Km

andi ≤ c + 1, · · · , (i, c) as N (2c−N+1)2 + N − i + 1 for c > Kmandi ≤ N, then we can write the

matrix PK as PK =  QK RK 0 1  ,

where QK is an L-by-L matrix giving the one-step transition probability between two transient

states, RKis an L-by-1 matrix giving the one-step transition probability from a transient state to the

absorbing statea, and 0 is the 1-by-L zero matrix.

The fundamental matrix associated with the absorbing Markov chain MCK is MK = (I −

QK)−1. LetmK(i, j) be the (i, j)-entry of MK. The matrix MKis an upper triangular matrix and,

unlike in the original protocol in Section 3, it can no longer be obtained in explicit form.

Once the matrix MK has been (numerically) computed the main QoS metrics can easily be

deduced, as shown below.

4.1

Expected delivery delay

We distinguish the casesK ≤ N − 1 and K ≥ N. In the former case, the expected delivery delay given that the chain starts in state(1, 0) reads

TK d = L1 X j=1 mK(1, j)E[Tj] = 1 µ K+1 X i=1 mK(1, L1− i + 1) iρ + i − 1 + K X i=1 PK−1 j=i−1mK(1, a(i, j)) N ρ + i − 1 ! ,

wherea(i, j) := 1 − i +(j+1)(j+2)2 , andTjis the sojourn time in the transient state with labelj.

IfK ≥ N we find TK d = 1 µ N X i=1 mK(1, L2− i + 1) iρ + i − 1 + N X i=1 PN −1

j=i−1mK(1, a(i, j)) +PK−1j=NmK(1, b(i, j))

N ρ + i − 1

! ,

whereb(i, j) := N − i + 1 +N (2j−N+1)2 .

4.2

Expected number of copies

The expected number of copies generated by the source before the delivery time, given that the chain starts in state(1, 0) is given by

GK d = K X c=1 c Km X i=1

P (absorption occurs in state (i, c)).

IfK ≤ N − 1 then GK d = ρ K−1 X c=1 c+1 X i=1 icmK(1, a(i, c)) N ρ + i − 1 + K K+1 X i=1 imK(1, a(i, K)) iρ + i − 1  ,

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while ifK ≥ N GK d = ρ N −1 X c=1 c+1 X i=1 icmK(1, a(i, c)) N ρ + i − 1 + K−1 X c=N N X i=1 icmK(1, b(i, c)) N ρ + i − 1 + K N X i=1 imK(1, b(i, K)) iρ + i − 1  ,

wherea(i, c) and b(i, c) are defined in Section 4.1.

The energy consumed by the source before the packet is delivered to the destination is given by

ptGKd while the energy consumed by all nodes during this period is(pt+ pd) GKd .

5

Asymptotic analysis

In this section we derive asymptotic results for the expected delivery delay and the expected number of copies at delivery instant in the two-hop relay protocol when the number of nodesN is large.

Deriving these asymptotic results from the explicit formulas in (1) and (2), respectively, is not easy (in the more simpler case when there are no timeouts getting asymptotics from the explicit results were already quite involved [8, Appendix A]).

We shall instead follow a mean field approach to find approximations of these asymptotics. The same approach was used in [14] and in [16] to derive asymptotic results for epidemic models.

The mean field approximation says thatX(t) (rep. G(t)), the expected number of copies (resp.

of copies generated by the source) in the network at timet, before absorption, when N is large, can

be approximated by the solution of the following 1st-order differential equations (see [12] for the general theory)

˙

X(t) = λ(N − X(t)) − µ(X(t) − 1), t > 0. (4)

˙

G(t) = λ(N − X(t)), t > 0. (5) The first equation simply reflects the fact that at timet X(t) increases with the rate λ(N − X(t)) and decreases with the rateµ(X(t) − 1). However the second equation reflects the fact that at time t

G(t) is a increasing function with rate λ(N − X(t)). We need to complement these equations with

another equation whose the solution approximatesD(t) := P (Td < t), the probability distribution

of the delivery delay. It was found in [14] that

˙

D(t) = λX(t)(1 − D(t)), t > 0. (6) Solving (4)-(6) with the initial conditionsX(0) = x0 (x0 = 1 in our model), G(0) = 0, and

D(0) = 0 yields X(t) = N λ + µ λ + µ +  x0− N λ + µ λ + µ  e−(λ+µ)t (7) G(t) = λN t − fN(t) D(t) = 1 − e−fN(t) (8) fN(t) = λ λ + µ h (N λ + µ)t +x0− N λ + µ λ + µ  (1 − e−(λ+µ)t)i (9)

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It can be checked thatD(0) = 0, limt→∞D(t) = 1 and t → D(t) is nondecreasing, so that D(t)

is indeed a probability distribution of a proper rv. As expected from the very definition ofX(t), we

note thatX(∞) = (Nλ + µ)/(λ + µ) is the expected stationary number of customers in a finite-state birth and death process, with birth rate (resp. death rate)λ(N − i) (resp. µ(i − 1)) in state

i ∈ {1, 2, . . . , N}.

5.1

Delivery delay

By definition,E[Td] =R0+∞P (Td> t)dt, so that from (7) E[Td] can be approximated by

E[Td] ≈

Z +∞

0

e−fN(t)dt

whenN is large. When N is large it is easily seen that the dominant contribution of e−fN(t)to

the above integral comes from small values oft since fN(t) is a nondecreasing function of N . We

distinguish between the following two cases ofx0: (1)x0is constant, (2)x0= αN with α ≤ 1.

x0constant: In this case,e−fN(t) can be approximated bye−f

00

N(0)t2/2 sincef

N(0) = 0 and

sincefN0 (0) = λx0does not depend onN , with f

00

N(0) = λ(N λ + µ − (λ + µ)x0). For 0 ≤ x0<

X(∞) this gives the 1st-order asymptotics E[Td] ≈ r π 2λ(N λ + µ − (λ + µ)x0) ≈ 1 λ r π 2N (N → ∞). (10)

The 2nd-order asymptotics ofE[Td] can be obtained by expanding fN(t) in Taylor series at the order

three in the vicinity oft = 0. We find E[Td] ≈ Z +∞ 0 e−f 00 N (0) 2! t 2 1 − f (3) N (0) 3! t 3dt = r π 2λ(N λ + µ − (λ + µ)x0) +(λ + µ)(N λ + µ − (λ + µ)x0) 3λ3(N − 1)2 (N → ∞).(11)

Figure 3 displays the 1st-order and 2nd-order asymptotics ofE[Td], given in (10) and in (11),

respectively, as a function ofN , and compare them with the exact value obtained in (1). We observe

that, asN increases, both asymptotics converge to the exact result.

x0 linear with N: In this casex0 = αN with α ≤ 1 is constant, fN(t) is approximated by

fN(t) ≈ f

0

N(0)t, since f

0

N(0) depend on N. So, this gives a 1st-order asymptotic

E[Td] ≈

1

λαN (N → ∞). (12)

The 2nd-order asymptotics ofE[Td] can be obtained by expanding fN(t) in Taylor series at the order

three in the vicinity oft = 0. We find E[Td] ≈ 1 αλN − λ − (λ + µ)α α3λ2N2 + (λ + µ)(λ − (λ + µ)α) α4λ3N3 (N → ∞) (13)

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20 40 60 80 100 1000 2000 3000 4000 5000 6000 7000 8000 9000 Number of nodes E[T d ] (s) 2nd−order asymp. Markov model 1st−order asymp. 20 40 60 80 100 500 1000 1500 2000 2500 Number of nodes E[T d ] (s) 2nd−order asymp. Markov model 1st−order asymp. (a) (b) 20 40 60 80 100 200 400 600 800 1000 1200 Number of nodes E[T d ] (s) 2nd−order asymp. Markov model 1st−order asymp. (c)

Figure 3: Comparing asymptotics for the expected delivery delay to the exact result (µ=0.001: (a) λ=0.0001, (b) λ=0.00025, (c) λ=0.0005.)

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Figures 4, and 5 displays the 1st-order and 2nd-order asymptotics ofE[Td], given in (12) and in (13),

respectively, as a function ofα, and compare them with the exact value obtained in (1). We observe

that, asN increases, both asymptotics converge to the exact result and that 2nd-order asymptotic is

closer to the exact value than the 1rt-order asymptotic.

0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 α E[T d ] (s) 2nd−order asump. Markov model 1st−order asump. 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 α E[T d ] (s) 2nd−order asump. Markov model 1st−order asump. (a) (b)

Figure 4: Comparing asymptotics for the expected delivery delay to the exact result as a function of

α (µ=1 and λ = 0.5: (a) N=50, (b) N=100) 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 α E[T d ] (s) 2nd−order asump. Markov model 1st−order asump. 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 α E[T d ] (s) 2nd−order asump. Markov model 1st−order asump. (a) (b)

Figure 5: Comparing asymptotics for the expected delivery delay to the exact result as a function of

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Asymptotics, asN is large, for any order moment of Tdcan be derived using a similar approach

as follows.

Lemma 1 WhenN is large, the nth-order moment of Tdis equal to

E[Tnd] E[Td] ≈ n! (λx0)n−1. (14) 

Proof: thenth-order moment of Tdis equal to

E[Tn d] = n Z +∞ 0 tn−1(1 − P (t))dt = λ Z +∞ 0 tnfN0 (t)e−fN(t)dt = λX(∞) Z +∞ 0 tne−fN(t)dt + λ(x 0− X(∞)) Z +∞ 0 tne−fN(t)−(λ+µ)tdt, (15)

where we integrate by parts and we used (9). WhenN is large the integral Z +∞ 0 tne−fN(t)−(λ+µ)tdt ≈ Z +∞ 0 tne−fN(t)dt.

So thenth-order moment of Tdbecomes equal to

E[Tn d] ≈ λx0 Z +∞ 0 tne−fN(t)dt = λx0 n + 1E[T n+1 d ], (16)

which by recurrence gives the Lemma. 

5.2

Expected number of copies at delivery instant

WhenN is large, E[Cd], the mean number of copies at the delivery time Td, is approximated by

R+∞

0 X(t)dD(t). With the use of (7) an integration by part gives

E[Cd] ≈ x0+ (N λ + µ − (λ + µ)x0)

Z ∞

0

e−fN(t)−(λ+µ)tdt (N → ∞).

By using again the property that the dominant contribution ofe−fN(t)−(λ+µ)t to the above integral

comes from small values oft. we may approximate e−fN(t)−(λ+µ)tbye−f 00 N(0)t 2/2 (Hintx0fixed). Hence, E[Cd] ≈ x0+r π 2λpNλ + µ − (λ + µ)x0≈ r πN 2 (N → ∞).

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5.3

Expected number of copies

WhenN is large, E[Gd], the mean number of copies generated by the source before delivery time

Td, is approximated byR0+∞G(t)dD(t). With the use of (8) an integration by part gives

E[Gd] ≈ λN Z +∞ 0 e−fN(t)dt − 1. Hence, E[Gd] ≈ r πN 2 (N → ∞), forx0fixed.

6

Validation and Numerical Results

In this section we first validate the Markov model introduced in Section 3 by comparing its perfor-mance (expected delivery delay) to that obtained by simulations, for two different mobility models (Random Waypoint (RWP) and Random Direction (RD) models). We then compare the expected delivery delay and the energy consumption induced by the two-hop relay protocol and the epidemic protocol. We conclude by investigating the performance of theK-limited two-hop relay protocol.

6.1

Model validation

We have simulated the two-hop relay protocol with exponential timeouts for both the RWP and the RD mobility models. In the RWP model [5] each node is assigned an initial location in a given area (typically a square) and travels at a constant speed to a destination chosen randomly in this area. The speed is chosen randomly in(vmin, vmax), independently of the initial location and destination.

After reaching the destination the node may pause for a random time, after which a new destination and speed are chosen, independently of previous speeds, destinations, and pause times. In the RD model [3] each node is assigned an initial direction, speed and travel time. The node then travels in that direction at the given speed and for the given duration. When the travel time has expired, the node may pause for a random time, after which a new direction, speed and travel time are chosen at random, independently of all previous directions, speeds and travel times. When a node reaches a boundary it is either reflected or the area wraps around so that the node reappears on the other side. In both mobility models nodes move independently of each other.

In our simulation settings, for both the RWP and the RD models the area is a square of side-lengthL = 2000m, the speed is constant and equals to V = 10m/sec., there is no pause time,

and the transmission rangeR is constant and the same for all nodes. In addition, in the RD model

the travel time is constant and equals to30sec. and the nodes reflect on reaching the boundaries. It

has been experimentally observed in [8] that wheneverR << L then the node inter-meeting time is

exponentially distributed with rateλ = 10.94RV

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For different values of the ratioR/L (resp. N ) , Table 1 (resp. 2) reports the expected delivery

delay obtained from the exact result in (1) and by simulations for the RWP model and the RD model, and give relative errors.

From the results in Tables 1-2 we conclude that, for both mobility models, the ratioR/L is a

key factor. More accurate results are obtained for the RD model for a given ratioR/L and a given

number of nodes (relative error of6% when the ratio R/L is less than or equal to 1.25% for 20 nodes

– see Table 2-A).

R/L (%) 1.25 1 0.5 0.1 1.25 1 0.5 0.1 Em[Td] (s) 1216 1529 3154 20102 1678 2116 4416 30264 Esim[Td] (s) 945 1245 2851 20861 1596 1988 4176 31651 |1 −ESim[Td] Em[Td] | (%) 22 18 10 4 6 6 5 4 (A) (B)

Table 1: Expected delivery delay calculated from (1) and by simulations (µ = 0.0001, N = 20: (A)

RWP model, (B) RD model). N 10 20 30 40 100 10 20 30 40 100 Em[Td] (s) 4344 3154 2596 2257 1436 6116 4416 3622 3141 1987 Esim[Td] (s) 4093 2851 2237 1839 1068 6208 4141 3297 2867 1512 |1 −ESim[Td] Em[Td] | (%) 6 9 14 18 26 1 6 9 9 24 (A) (B)

Table 2: Expected delivery delay calculated from (1) and by simulations (µ = 0.0001, R/L = 0.5%:

(A) RWP model, (B) RD model).

6.2

Comparison of two-hop relay and epidemic routing protocols

In this section, we compare the expected delivery delay,E[Td], and the expected number of packets

transmitted,E[Gd], as a function of µ, the timeout intensity, for the two-hop relay and the epidemic

routing protocols. The absorbing Markov chain modeling the epidemic routing protocol is the same as the absorbing Markov chain in Section 3, except that the birth rate in statei is now equal to λi(N − i), since in the epidemic routing protocol all nodes are allowed to generate copies of the

packet. The death rate (resp. absorption rate ) in statei is unchanged and equal to µ(i−1) (resp. λi). The computation of the expected delivery delay and the expected number of packets transmitted for the epidemic routing protocol is therefore similar to that carried out for the two-hop relay protocol, except that the fundamental matrix for the epidemic routing protocol cannot be computed in explicit form. This matrix was obtained numerically.

As expected, we observe that the epidemic routing protocol induces a smaller expected delivery delay than the two-hop relay protocol, but at the expense of a much more important overhead in terms of the number of copies generated. We also point out that the conclusions drawn from the results in Figure 6b apply for the energy consumptionsPsandPdin the case where the energy to

transmit (resp. decode) a packet is constant, since we have shown in Section 4 that in this casePs

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0 0.02 0.04 0.06 0.08 0.1 0 100 200 300 400 500 600 µ E[T d ] (s) λ=0.001, 2−hop λ=0.001, epidemic λ=0.005, 2−hop λ=0.005, epidemic 0 0.02 0.04 0.06 0.08 0.1 0 20 40 60 80 100 120 µ E[G d ] λ=0.001, 2−hop λ=0.001, epidemic λ=0.005, 2−hop λ=0.005, epidemic (a) (b)

Figure 6: Expected delivery delay (left) and expected number of packet transmitted (right) for two-hop relay and epidemic routing protocols as a function ofµ (N = 100).

6.3

Limited energy consumption

For different values ofλ, the inter-meeting time rate, Figure 7a plots the expected delivery time, TK

d , under theK-limited two-hop relay protocol for different values of K, the maximum number

of copies of the packet that the source may generate (see Section 4). For eachλ, we observe there

exists a thresholdK0such thatTdK is almost constant whenK ≥ K0(K0 ∼ 20 for λ = 0.001).

Figure 7b plots the expected delivery time,GK

d , under theK-limited two-hop relay protocol for

different values ofK. Similarly to TK

d , We observe that there exists a thresholdK0such thatTdKis

almost constant whenK ≥ K0

7

Concluding Remarks

In this work, we have evaluated the main QoS metrics of the two-hop relay protocol under the assumption that packets in relay nodes have a limited lifetime. Closed-form expressions have been derived for the probability distribution of the packet delivery delay, the expected number of copies in the system at the delivery instant, and the overall expected number of copies generated by the source at the delivery instant. We have observed that the latter metrics is directly related to the energy needed to transmit the packet to the destination node, in the case when the energy needed to transmit a packet between two nodes and the energy needed to decode a packet are constant. We have also proposed, and evaluated, a modification of the two-hop relay protocol that limits the number of copies of the packet that the source may generate.

In this paper our work has focused on the performance of the two-hop relay protocol before the destination receives the packet for the first time. It would also be interesting to quantify the impact of using an anti-packet mechanism on the total amount of energy consumed by the network during

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10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 K

Expected Delivery Delay (s)

λ=0.001 λ=0.005 λ=0.01 10 20 30 40 50 60 70 80 90 4 5 6 7 8 9 10 11 12 K

Expected number of copies λ=0.001 λ=0.005 λ=0.01

(a) (b)

Figure 7: Expected delivery delay (left) and expected number of packet transmitted (right) under

K-limited two-hop relay protocol for different values of K (N =100, µ=0.001).

the entire lifetime of the packet, including its copies, in the network. Also, we have assumed that there is no timeout on the packet lifetime at the source. This assumption may not be realistic in some applications, and would therefore be worthwhile to relax it.

This study is part of a research effort towards developing simple analytical models for quantifying the performance of relay protocols for MANETs and, in particular, for better understanding the delay-energy tradeoff of this class of protocols.

Appendix I: The fundamental matrix

Lemma 2 The matrix I-Q hasN distinct, real, and strictly negative eigenvalues z1, . . . , zN given

by

zk= −N(2ρ + 1) + 1 − (N + 1 − 2k)

√ 4ρ + 1

2 , k = 1, 2, . . . , N.

Therefore the fundamental matrix M= (I − Q)−1exists, and its (i,j)-entry is given by

m(i, j) = −N ρ + j − 1N −1 i−1ρi−1 N X k=1 Ψk iΨkj zkΨkτ2(Ψk)T , (17)

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with Ψk= (Ψk 1, . . . , ΨkN) where Ψki = min(i−1,k−1) X l=max(0,i−1−N+k) k − 1 l  N − k i − 1 − l  (−1)i−1xk−1−l1 xN −k−i+1+l2 , x1 = −1 − √ 1 + 4ρ 2ρ , x2= −1 +√1 + 4ρ 2ρ ,

and whereτ = diag (τ1, . . . , τN), with τi =



N −1 i−1ρi−1

−1/2

. 

Proof. To simplify the computation of M we introduce the matrix A defined as

A= −B(I − Q), (18) where B= diag (b(1), . . . , b(N )) with b(i) = N ρ + (i − 1). Matrices M are A are related through the simple identity M = −A−1B. In the following we will compute A−1. We will follow the

approach developed in [1]. We first compute the eigenvalues and left/right eigenvectors of A.

Eigenvalues of A.

Letz be some eigenvalue of A and let Ψ = (Ψ1, . . . , ΨN) be the associated left eigenvector.

That is, ΨA= zΨ, or equivalently,

ρ(N − (i − 1))Ψi−1− (ρN + i − 1 + z)Ψi+ iΨi+1= 0 (19)

fori = 1, . . . , N , with Ψ0 = ΨN +1 = 0 by convention. Let ψ(x) = PNj=1Ψjxj denote the

generating function of Ψ. Multiplying (19) byxiand then summing overi yields

ψ0(x)

ψ(x) =

ρN x − (ρN − 1 + z) − 1/x

ρx2+ x − 1 . (20)

Let the zeros ofx2+ x/ρ − 1/ρ be x1= −1− √

1+4ρ

2ρ and x2=−1+ √

1+4ρ

2ρ . The unique solution

of (20) such thatΨN= 1 is ψ(x) = x (x1− x)c1(x2− x)c2 (21) c1:= x 2 1ρN − x1(ρN − 1 + z) − 1 ρx1(x1− x2) , c2:=−x 2 2ρN + x2(ρN − 1 + z) + 1 ρx2(x1− x2) .

It is easily seen thatc1+ c2= N − 1 (Hint: use x1x2= −1/ρ), so that (21) also writes

ψ(x) = x (x1− x)c1(x2− x)N −1−c1. (22)

Becauseψ(x) is a polynomial of degree N , we observe from (22) that necessarily c1is an integer

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The equationsc1= k − 1 for k = 1, . . . , N give the following N eigenvalues of A:

zk = −N(2ρ + 1) + 1 − (N + 1 − 2k)

√ 4ρ + 1

2 , k = 1, . . . , N. (23)

All eigenvalues of A are distinct (obvious from (23)). Furthermore,zkincreases ask increases, and

it is easily seen thatzN< 0 for ρ > 0. Thus, zk < 0 for all k = 1, · · · , N.

Left eigenvectors of A.

Recall that Ψk = (Ψk

1, . . . , ΨkN) is the left eigenvector associated with the eigenvalue zk of

A. Theith component Ψk

i of the eigenvector Ψk is the coefficient ofxi in the polynomialx(x1−

x)k−1(x 2− x)N −kthat is Ψki = (−1)i−1xk−11 xN −k−i+12 min(i−1,k−1) X l=max(0,i−1−N+k) k − 1 l  N − k i − 1 − l   x2 x1 l . Right eigenvectors of A. Recall that Φk = (Φk

1, . . . , ΦkN)T is the right eigenvector associated with the eigenvaluezk,

fork = 1, . . . , N . We proceed like in [1, Section 2.4], that is we look for a diagonal matrix τ =

diag(τ1, . . . , τN) such that

τ−1Aτ = τ−1T

. (24)

It is easily found that (Hint: solveτ2

i/τi+12 = ρ(N − i)/i for i = 1, . . . , N with τ1= 1)

τi= N i  i Nρ i−1 −1/2 , i = 1, . . . , N

satisfy (24). The identity ΨkA= zkΨk implies that Ψkτ (τ−1Aτ ) = zkΨkτ . Therefore, Ψkτ is

a left eigenvector of the matrixτ−1Aτ associated with the eigenvalue zk. Since the matrixτ−1

is symmetric, it has identical left and right eigenvectors. Hence,(τ−1Aτ )(Ψkτ )T = z

k(Ψkτ )T

which gives that Aτ2k)T = z

kτ2(Ψk)T. This shows thatαkτ2(Ψk)T is a right eigenvector

associated with the eigenvaluezkfor any constantαk 6= 0.

Without loss of generality we select the constantsα1, · · · , αNso that ΨkΦk= 1 for every k =

1, · · · , N. Hence, αk = 1/Ψkτ2(Ψk)T fork = 1, . . . , N . Finally, Φk= τ2(Ψk)T/Ψkτ2(Ψk)T,

or equivalently Φki = 1 Ψkτ2k)T N i  i Nρ i−1 −1 Ψki, i = 1, . . . , N. (25)

The proof is concluded by noting that

ˆ a(i, j) = N X k=1 Φk iΨkj zk =N − 1 i − 1  ρi−1 N X k=1 Ψk iΨkj zkΨkτ2(Ψk)T , (26)

by using (25). Equation 26 together withm(i, j) = −(Nρ + (j − 1))ˆa(i, j) (coming from M =

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Remark 7.1 The solution to (20) is not unique, since if ψ(x) is a solution then αψ(x) is also a

solution for any constantα 6= 0. This is of course related to the fact that eigenvectors are uniquely defined up to a multiplicative constant. Takingα = 1 (like we implicitly did in (22)) implies that the N th component, ΨN, of the left eigenvector Ψ is taken to be equal to1 (coefficient of xN in (22)).

Remark 7.2 Replacingx by 1 in (20) implies the following relation between the eigenvalue zkand

its corresponding left eigenvector Ψk N X j=1 jΨkj = − zk ρ N X j=1 Ψkj, (27)

that is true for1 ≤ k ≤ N.

Appendix II: Distribution of delivery delay

The delivery delay,Td, given that there arei copies in the network at time 0 is the time to absorption

of the Markov chain MC of Figure 1, given thatI(0) = i. In order to compute the distribution of Td,Pi(Td ≤ t), given that I(0) = i, first we derive the Laplace Steiltjes Tansform (LST) given that

I(0) = i, fi(s) = Ei[e−sTd], and next we invert fi(s). fi(s) will actually hold for any complex

numbers such that <(s) ≥ 0. Starting at state i, if we condition on the next possible transition of the MC,fi(s) reads fi(s) = λi λN + µ(i − 1) + s + µ(i − 1) λN + µ(i − 1) + sfi−1(s) + λ(N − i) λN + µ(i − 1) + sfi+1(s) (28)

fori = 1, . . . , N (by convention f0(·) = fN +1(·) = 1).

Multiplying both sides of (28) byλN + µ(i − 1) + s and dividing by µ, yields

(i − 1)fi−1(s) −



ρN + i − 1 +µs 

fi(s) + ρ(N − i)fi+1(s) = −ρi (29)

fori = 1, . . . , N , with ρ := λ/µ.

Let f = (f1(s), . . . , fN(s))T.

In matrix form (29) writes

 A s

µI 

f = b, s ≥ 0, (30) with I theN -by-N identity matrix, b := (−ρ, −2ρ, · · · , −ρN)T, and A is the N-by-N matrix defined in (18) in Appendix I. It is shown in Appendix I that the matrix A is invertible and diago-nalizable, namely, there exists an invertible matrix F such that

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wherez1, · · · , zNare the eigenvalues of A, the jth right eigenvector of A, Φj, is thejth column of

the matrix F, and the left eigenvector, Ψi, is theith row of the matrix F−1. Hence

A− (s/µ)I = F diag (z1− s/µ, · · · , zN− s/µ) F−1,

and

(A − (s/µ)I)−1= F (diag (1/(z1− s/µ), · · · , 1/(zN− s/µ)) F−1 (31)

provided thatzk−s/µ 6= 0 for k = 1, · · · , N. Since, we know from Appendix I, that the eigenvalues

of A are all strictly negative, so that the right-hand side of (31) is well defined (in particular) for all

s ≥ 0. Therefore, (cf. (30))

f = F (diag (1/(z1− s/µ), · · · , 1/(zN− s/µ)) F−1b, s ≥ 0. (32)

Since Φj is thejth column of the matrix F, and Ψiis theith row of the matrix F−1, we see from (32) that theith component, fi(s), of the vector f is given by

fi(s) = −ρ N X j=1 N X k=1 jΦk iΨkj zk− s/µ , s ≥ 0. (33)

Closed-form expressions for the eigenvalues and right/left eigenvectors of the matrix A are provided in Lemma 2. Lemma 2 and Remark 7.2 gives that

fi(s) = N i  i Nρ i−1−1 NX k=1 Ψk i(Ψk1T) Ψkτ2k)T zk zk− s/µ s ≥ 0, (34)

where 1T is the column vector of dimensionN whose all components are equal to 1, one.

Equa-tion 34 implies that the distribuEqua-tion ofTdgiven that the MC starts at state1 reads

Pi(Td< t) = 1 −N − 1 i − 1  ρi−1 −1 N X k=1 Ψk i Ψkτ2k)TΨ k1Tezkµt. (35)

Sincezk < 0 for 1 ≤ k ≤ N and ρ > 0, thus limt→+∞Pi(Td > t) = 0. Further, Pi(Td> 0) =

1 (Hint:ΨiΦj

= 0 for i 6= j, and ΨkΦk= 1).

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[1] D. Anick, D. Mitra and M. M. Sondhi, Stochastic Theory of a Data-Handling System with Multiple Sources, Bell System Technical Journal, Vol. 61, No. 8, October 1982, pp. 1871-1896. [2] F. Baccelli and P. Brémaud, Palm Probabilities and Stationary Queues. Springer-Verlag, 1987. [3] C. Bettstetter, Mobility Modeling in Wireless Networks: Categorization, Smooth Movement, Border Effects, ACM Mobile Computing and Communications Review, 5, 3, pp. 55-67, July 2001.

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of Mobile Networks, Vol. 10, No. 5, pp. 555-567, Sept 2004.

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[6] A. El Gamal, J. Mammen, B. Prabhakar and D. Shah, Throughput-Delay Trade-off in Wireless Networks, Proc. of INFOCOM 2004, Hong Kong, 2004.

[7] C. Grinstead and J. Snell, Introduction to Probability. American Mathematical Society, 1997. [8] R. Groenevelt, P. Nain and G. Koole, The Message Delay in Mobile Ad Hoc Networks, Proc. of

Performance 2005, Juan-les-Pins, France, October 2005. Published in Performance Evaluation,

Vol. 62, Issues 1-4, October 2005, pp. 210-228.

[9] M. Grossglauser and D. Tse, Mobility Increases the Capacity of Ad hoc Wireless Networks,

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Information Theory, Vol. 46, No. 2, March, 2000, pp. 388-404.

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[12] T. G. Kurtz, Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes, J. Applied Prob., Vol. 7, pp. 49-58, 1970.

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INFOCOM 2005, Miami, FL, USA, March 2005.

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Figure

Figure 1: Transition rate diagram of the Markov chain { I (t), t ≥ 0 } .
Figure 2: The modified absorbing Markov chain with N absorbing states.
Figure 3: Comparing asymptotics for the expected delivery delay to the exact result (µ=0.001: (a) λ=0.0001, (b) λ=0.00025, (c) λ=0.0005.)
Figure 4: Comparing asymptotics for the expected delivery delay to the exact result as a function of α (µ=1 and λ = 0.5: (a) N=50, (b) N=100) 0.2 0.4 0.6 0.8 100.020.040.060.080.10.120.140.160.180.2 αE[Td] (s) 2nd−order asump.Markov model1st−order asump
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